Download Stability of extreme value for a multidimensional sample

Statistique et analyse
des données
M ARIE -F RANÇOISE D ELCROIX
P IERRE JACOB
Stability of extreme value for a multidimensional sample
Statistique et analyse des données, tome 16, no 1 (1991), p. 1-21
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Statistique et Analyse des Données
1991 - Vol. 16 n° 1 p. 1 - 21
STABILITY OF EXTREME VALUE FOR
A MULTIDIMENSIONAL SAMPLE
Marie-Françoise DELCROIX and Pieme JACOB
Laboratoire de Statistique et Probabilités (M2)
Université des Sciences et Techniques de Lille Flandres-Artois
59655 VILLENEUVE D'ASCQ CEDEX (France)
Abstract.
Let (Y lv „,Y n ) be a random samplefront a continuous distribution function F
E + . If Yfl dénotes îhe highest value ofthis sample, thenthe highest value of
*
F(Y 1 ),...,F(Y n ) is F(Y n ). This simple remark suggests a natural définition for the
over
highestvalue Xn = (Rn,©n) of a random sample
( X ^ . . . ^ ) in R k , based upon
the polar représentation ^ , 0 ^ , . . . , ( 1 ^ , 0 ^ of thèse variables. Precisely, if F e is the
conditional distribution function of R given 0 = 0, we define the maximum value of
the sample as the observation X* which maximizes Fg(R). This définition is attractive
because it is not based only on a classical distance in R k , but, which seems more
relevant, on the probability to be at a certain distance from the origin. This notion allows
us to study the stability ofsuch extrême values. Of course, a lot of multidimensional
distributions do not hâve stability properties. So we need a weaker notion than stability to
go on. The idea is to substitute a variable Xq> = (<p(R),0) for each observation
X = (R,0), where <p is a suitable function, in order to obtain stability properties for
the variable
X(p. It consists in considering a new set of points
E^ = {((p(R1)>91),...,(cp(Rn),0n)} instead ofthe initial sample. As shown in thispaper,
the function q> must be sufficiently concave.
Key-words: sample, isobar, extrême value, stability, relative stability, asymptotic
localization.
A.M.S. Subject classification : 62 G 30, 60 D 05
Manuscrit reçu le 12 juin 1991, révisé le 10 février 1992.
2
M.F. Delcroix et P. Jacob
I. INTRODUCTION
Nowadays, theory of extrême values concerns often non identically distributed
data, dépendent data ([Haiman, Puri, 1990],[Haiman, Puri]) or multivariate independent
identically distributed data ([Davis, Mulrow, Resnick, 1987]). However récent papers
about outliers ([Gather, Rauhut, 1990],[Green, 1976],[Mathar, 1989],[Munoz-Garcia,
Moreno-Rebollo, Pascual-Acosta, 1990]), give a new interest to the old notion of
stability ([Geffroy, 1958,1959],[Geffroy, 1961],[Gnedenko, 1943]). We propose hère a
new définition for the highest value of a multidimensional sample and for the stability of
this highest value. It is also possible to define outlier-resistant or outlier-prone
distributions as it has been done for Ek-valued variables in [Gather, Rauhut, 1990],
[Green, 1976]. However, in the first step of this study, we examine the properties of
such extrême values : in the présent paper we focus our attention on the stability of the
extrême value of a sample.
In this paper we consider random variables defined on a probability space
(Q,ClJP) and with values in the Euclidian space IRk.
Forevery x in R*\{0} we define a pair (llxll,^) = (r,8) in R + * xS*"1, where
11.11 is the Euclidean norm. The unit sphère Sk_1 in R k is endowed with the induced
topologyof E k .
For each random variable X = (R,0), we assume that the distribution of 0, and
for ail 8, the distribution of R given 0 = 6, hâve a continuous density. We name F e
the continuous conditional distribution function of R, given 0 = 8, and F^ its
generalized inverse. For each 0 < u < 1, we name u-level isobar - from the distribution
of R given 0 = 8 - the mapping 8 —» F"e (u). We suppose that this mapping is
continuous and strictly positive ; the surface which équation is p = F^ (u) is also named
an isobar.
Let En = (X ls ...,X n ) be a sample of independent random variables with the
same distribution as X. For each 1 < k < n there is almost surely an unique uk-level
isobar from the distribution of R given 0 = 8 which contains (Rk,0k). We define the
*
* *
maximum value in En as the point X n = (R n ,0 n ) which belongs to the upper level
isobar, i.e. the isobar which level is max uk . Obviously, we are not able to find this
l<k<n
maximum value of a sample from an unknown distribution, whereas it can be done with
the farest pointfromthe origin or with thefictitiouspoint having the largest coordinates
of the sample. However this kind of extrême value and, more generally, the extrême
STABILITY OF EXTREME VALUE
3
values obtained by ordering the sample according to the levels, hold more information on
the conditional distributions tails and allow a statistic survey of the isobars.
* *
In section II, we specify some properties of the distribution of the pair (Rn>0n).
In section III and IV, we define the notions of stability in probability and almost sure
stability of the maximum value. Roughly the idea lying back of the définition is the
*
*
tendency of Xn to be near a given surface. More precisely, X n is called stable in
probability (or almost surely) if there is a séquence (r n ) of surfaces, which équations are
*
*
p = g n (8), such that Rn - g n (0 n ) -> 0 in probability (or almost surely). For a class of
distributions we précise, this phenomenon occurs and (gn) turns out to be a séquence of
isobars. This expands the notion of stability studied by J. Geffroy in [Geffroy,
1958,1959]. Actually we use several of his methods. Examples are given in section V. In
section VI, the assumptions we hâve done throughout this paper are discussed, especially
some regularity conditions for the isobars. At length, in section VII we give some
properties and examples about (p-stability.
II - PRELIMINARIES
*
In this section we give some results about the conditional distribution of Rn
given 0 n = 8. They will be used in the sequel of this paper.
Let (X lv ..,X n ) be a sample with polar représentation (R 1 ,6i),...,(R n ,0 n ). For
each 1 < i < n, put :
(1)
E-tFeXRj) = maxF e (R j )}.
1
j=l
J
For aïl 8 and for ail 0 < t < 1, P(F e (R) < t/0 = 8) = Fe(F"e1(t)) = t, hence
{Fe.(Rj). j = l».»»n} is a sample from the uniform distribution over [0,1]. Now the
maximum value of the sample is almost surely defined as the point X n which polar
représentation is :
<K*b=£"=i(Ri'ei>v
4
M.F. Delcroix et P. Jacob
PROPERTYI.-
*
a) 0 n and 0 are identically distributed.
b)Any u-level isobarfromthe distribution of R given 0 isalso
*
*
the xx ~level isobar from the distribution of Rn given 0 n .
n
i
a) Since P(F e (R) < t/0 = 8) = t, for each l < j < n
0j and F e .(Rj) are
independent. It follows easily that {0; ; j = l,...,n} and {Fe.(R:) ; j = 1
n} are
independent.
Thus, for each 1 < j < n, 0 : and 1 E . are independent. Consequently, for any
Borel set C of S*"1 :
(2)
P ( 0 > C) = P ( Z 0 i l E L e C ) = E P ( 0 i € C;Ei)
i=l
i=l
n
= X P(0 f e C) P(Ei) = P(0 e C)
b) Let p = FQ (U) be an u-level isobar from the distribution of R given 0 = 9
andlet B be the event {R* < F:i(u)}. Since B = fl"
wn
*
{F e .(Ri) £ u}, B is
1
independent of {0:, j = l,...,n}. Thus for any Borel set C of Sk_1 (2) implies :
n
n
P ( 0 * e C ; B ) = X P ( 0 : e C ; E : ; B ) = E P(©i e C) P(Ei;B)
n
i=l
i=l
= P(0*e C)P(B)
*
Thus © n and 1 B are independent ; therefore,
(3)
n
P(R! < F i ( u ) / ®* = 6) = P(B) = FI P(F e (R^ <ï u) = U* .
t»n
i=l
i
l
*
COROLLARYL0 * = 8. For any
Let F n 9
8, F * Q = F £ .
*
be the conditional distribution fonction of 1^ given
STABILITY OF EXTREME VALUE
i Let S be the support of the distribution of X. Let x = (r,8) be a point which distance
from the nearest isobar is strictly positive. Taking account of the isobar's continuity,
there exists an open bail B(x,e) which distance from the nearest isobar is also strictly
positive. Therefore the distribution of X assigns a null mass to B(x,e). But the support
S is the set of ail points z in R k such that P(V) > 0 for each open set V containing z ;
hence x is not a point of 5. Thus the distance between any point of S and an isobar is
zéro. In the sequel of the proof, we shall consider as an isobar any (uniform) limit of a
decreasing séquence of isobars. S is then the union of ail the isobars from the
distribution of R given 0 = 8. For any open set 0 in S**1 and for any pair (g,h) of
isobars such that g < h, define
D(0,g,h) = {x = (r,8) e S : 8 E 0, g(8) < r < h(8)}.
Clearly the class *Ll of thèse sets is a K-system [Billingsley, 1968].
Moreover for ail x in S and for ail e > 0 there is a set D in *U, with diameter less
o
o
than e, such that x e D C D (D dénotes the interior of D for the induced topology on
S). By [Billingsley, 1968] page 14, *U is a determining class for the separable metric
space S.
Let X = (R,0) be a E*-valued random variable such that 0 and 0 are
identically distributed and such that the distribution function of R given 0 = 8 is FJJ. It
suffices now to show that X and X* are identically distributed in order to obtain the
corollary 1.
This follows immediady from (3),fromthe following équation :
(4)
P(R < g(©)/0 = 8) = F£(g(8)) = u n ,
andfromthe fact *U is a determining class.
The previous results state that both the distribution of R given 0 and the
distribution of R* given 0 hâve the same set of isobars. Hence we will deal only with
n °
n
the formers. In the sequel any u-level isobar from the distribution of R given 0 is
labelled as u-level isobar.
We assume in the remainder of this paper that for ail 8 the mapping F e is
strictly increasing and thus bijective. Fix a point 82 in Sk"! and provide the axis (08^
with the unit vector 08! . For every point w on the positive half axis 08j , there is an
6
M.F. Delcroîx et P. Jacob
unique isobar containing w, which level is denoted by u(w). Let p = g(8,w) be the
équation of this u(w)-level isobar (note that g(8j,w) = w). Moreover the mapping
w ~-> u(w) from R+* into]0,l[ is increasing and bijective. The following condition
(H) will be used in most theorems.
(H)
there exists 0 < ax < Pj < + °° such that for ail 8 in Sk*l
andforall w>0:oc! < | ^ (9,w) < pj
An immédiate conséquence is given by the next property :
2.- For ail e > 0, there exists t\ > 0, and for ail w > 0, there exists
two isobars h (8,w) and he(8,w) such that for every 8 :
PROPERTY
e
(5)
g(8,w) - e < he(8,w) < g(8,w) - TI < g(8,w) + Ti < h£(8,w) < g(8,w) + e
(we lay stress on the fact that T| does not dépend upon w).
i By the mean value theorem, we obtain :
and
g(8,w) + Jtax < g(8,w + i ) < g(8,w) + Â$x
g(8,w) + i p 1 < g(8,w + i ) < g(8,w) + £ax
if i > 0
ifi<0
£
It suffices to choose i = — (resp. Jt = -e/p^ and to put
Pi
(6)
Ti = e a 1 / p 1
(7)
he(8,w) = gîS.w + e/p!)
(8)
he(8,w) = g(8,w-e/p 1 ).
i
REMARK
1.- The level u(w + e/p^ (resp. u(w - e/p^) of he(8,w) (resp. of
fie(8,w)) is a increasing function of w or of u(w).
REMARK 2.-
Actually, (5) is a key-property, but H is somewhat easier to handle.
H was suggested to us by an unpublished work of Geffroy on a closely related topic.
Most détails can be found in the thesis of Lecoutre [Lecoutre, 1982]. Geffroy considered
STABILITY OF EXTREME VALUE
7
the case of an unimodal density in R 2 , decreasing in any direction. The sample was
ordered according to the level-lines of the bivariate density of the isobars. Anyway, (H)
is not necessary as shown in section VI.
REMARK 3.-
For a gaussian sample of R2-vectors with covariance matrix f
1,
the hypothesis (H) is satisfied. We point out that the isobars are also the level-lines
of the bivariate density. Their polar équations are g(6,w) = w<p(8) where
mfltt
1
fcos 2 0 , sin29 \-l/2
m - STABILITY IN PROBABILITY OF X* = (R*,0*).
By property 1, the distributions of (R£,©„) and of (R,0) do hâve the same set
of isobars. So we can propose the following définition.
1. DÉFINITION 1.- (XjJ)n is stable in probability if and only if there is a séquence
(gn)n » of isobars satisfying
(9)
R^-g^^O.
P
2. PROPOSITION 1.- Suppose that (H) holds. If R* - gn(©n) -> 0 (where (gn)n
is a séquence of isobars), thenfor ail e > 0 :
(10)
minP(R£ < gn(8) + e/©* = 8 ) - » l and
e
(1D
max P(R* < gn(8) - e/Q*n = 8) -> 0
e
i Let e > 0 ; by property 2, there is T| > 0 and two séquences of isobars (h^)n and
(îij;)n such that for ail 8 and for ail n,
(12)
g n (8)-e < fij(9) < g n (8)-Ti <g n (8) + îl< h£(8) < gn(8) + e ,
It follows that for eachfixed8 :
(R^<g„(e)+Ti) c (R;<hS(e)).
M.F. Delcroix et P. Jacob
Since R£ - g n (0*) ^ 0 ,
(13)
I i m P(R; < h^(0*)) = 1.
n—H-*»
Furthermore, if v £J1 is the level of h£ :
(14)
P(R* < h ^ ) ) = J ^ x P(R*n < h^n(8)/©* = 8) Pe; (d8)
=
J S k-i v ^ p e: ( d e ) = v ^
And for every 8, v £ ^ = P(R*n < ^(8)/©* = 8) < P(R^ < g n (8) + e/©* = 8).
We deduce (10) from (13) and (14). The proof of (11) can be treated in the same way. i
This proposition provides a criterion of stability in probability.
3. THEOREM 1.- Let (gi)n be a séquence of isobars ;
ï)if (R)holdsandif
(15)
lim
n_>00
R * - g n ( © n ) - > 0 , Xhenforall
0<a<b<l:
sup [(F; 0 )-l(b)-(F; 9 )-l(a)]=O
9€S k_1
ii) conversely, if (15) holds, then X* = (R*,©*) is stable in probability.
i Let e >0. In view of the proposition 1, for n large enough, b<P(R*<g n (8)+e/0* = 8)
for any 8 .
Then for every 8 , ( F * ^ ) " 1 * ) < g n ( 8 ) + e . Similarly,
( F n e ^ " 1 ^ > g n< e > " e f o r a n y 9* T h u s <15> f o l l o w s - Conversely, for e > 0,
(F* e )-i(l-e) - (F* e)_1(e) converges to 0 uniformly. We choose by a diagonal method a
séquence (en) which decreases to 0, such that :
converges to 0 uniformly
Putting V e ) = ( ¾ . ^ 1 - ¾ )
Bnd
g„(6) = (F;9)^(¾).itfoUowsthat:
P(R; <h„<e)ie;; = e) = 1~*a
and
STABILITY OF EXTREME VALUE
P(R; <g n (0)ie; = e) = en for ail e ,
sothat
P
(gn(®^X^<hn(©>> 1-26».
This complètes the proof of the theorem 1.
The corollary 1 gives for each 0 < y < 1 : (Fj| ^ ( y ) = ^(yl,n)
i
([Geffroy,
1958,1959], page 70) and this entails the next theorem. The proof is the same as for
theorem 20 in [Geffroy, 1958, 1959].
4. THEOREM 2 . -
i ) / / (H) holdsandif R* - gn(0*) ^ 0 , thenforail0 < a< P :
(16)
Lim
sup
I F'Ml - o/n) - F ^ l - p/n) I = 0.
ii) If ( 16) holds then (X*)n is stable in probability.
REMARK 4 . -
a)F^(l-p/n) and F^l-a/n) are meaningful for p/n<l and for oc/n < 1.
b) For ail 8 and for ail 0 < t < 1, put G^(0 = FQ (1-0 (where Ge = 1 - F e ).
We hâve just proved that if (X*)n is stable in probability, the "highest" values of the
sample are then stable "about" the isobars yn(8) = G"^-), for n large enough.
5. Gnedenko's theorem ([Geffroy, 1958,1959],[Gnedenko, 1943]) gives a
simple criterion of stability in Probability in our context.
THEOREM 3.-
Suppose that (H) holds, then (X*)n is stable in Probability if and
onlyiffor some Qx :
G9lW
(17)
lim ^—j—rr =0 , for ail h > 0
x->+oo ^ e ^ x " n ^
(Then (17) is truefor ail Q).
10
M.F. Delcroix et P. Jacob
Put W* the intersection of 06"J" with the isobar containing X* (see section H).
For ail w > 0 ,
P(W* < w) = P(R*<g(0*,w)),
where g is the isobar which équation is g(6,w).
Since g is an isobar, P(R* < g(6,w) / 0 * =8) does not dépend upon 8, then
P(W*<w) = P(R* < g ( 8 1 , w ) l 0 * - 8 1 )
= P(R*< w I 0 ^ = 8 ^
=
%&)•
Let W| be the intersection of 06"}" with the isobar containing X^ - (R i ,0 i ), (i=l,...,n).
For ail w > 0,
(18)
P(Wi<w) = P C R j ^ w / e ^ e ^ ^ F e ^ w ) .
The random variable W* is then the maximum of n E + -valued i.i.d. variables,
W 1 ,..,W n , having a common distribution function F 0 . From Gnedenko's theorem ,
(W*)n is stable in Probability if on only if :
O e i (x)
l i m ^ / u\ = 0 , for ail h > 0
x_>+oo G Gl (x-h)
Now it suffices to prove that (W*)n is stable in probability if and only if (X*)n is stable
in probability.
Let e > 0, suppose that (W*)n is stable in Probability ; there exists a séquence
P
~e e
( a ^ such that W * - a n ->0 .Let Jtn be the isobar containing ^ and h n , hn the
isobars satisfying (5) for ail 8 :
(19) i n (8) - e < h£(9) < i n (8) (T| does not dépend upon n).
Then,
TI < i n (6)
< i n (6) + TJ < h^(8) < i n (6) + e
STABILITY OF EXTREME VALUE
(20)
11
{iWJ-aJSTi} C { i f t e ^ R * <î h^(0*)} C
*Z-*n(@t>
sothat
{\R*-An(@*)\<e}
-^°
Conversely, if there exists a séquence of isobars (g„) n such that
P
+
R*-g n (0*) -»0,note a„ the intersection of 00 1 with g n . For e > 0, there exists
T] > 0 and for ail n, there exists hjj and hjj satisfying (19) for ail 0 ; then,
(21) {lR*-gn(0;)l<Ti} C {h^(6*)<R* < l#e*)} c f l w j - a j s e }
and this achieves the proof.
IV - ALMOST SURE STABILITY OF (X£) n .
1.- DEFINITIONS.
a) DÉFINITION 2.- The séquence (X*)n is almost surely stable if and only if there
is a séquence (gn)n of isobars such that :
(22)
RÎ-EnCQS)^0-
By theorem 2, if (X*) is almost surely stable, yn(8) = G ' ^ ) is a convenient gnsequence. Thus, in the remainder, Yn(8) will dénote the isobar G'B (-) and T n the set of
points {(p,8);p<Y n (e)h (n * 2).
For e > 0, put :
(23)
r^={(p,8):p<Y n (8) + e}
(24)
r-£={(p,e):p<Yn(8)-e}
12
M.F. Delcroix et P. Jacob
Then, (X*)n is almost surely stable if and only if for ail e > 0,
P{Liminf(X*e i j - r " e ) } = 1
(25)
2.- As it has been done with Gnedenko's theorem, it is possible to prove the next
theorem, with theorems 49 and 50 of [Geffroy, 1958,1959],
THEOREM 4.-
If (H) holds, then (X*)n is almost surely stable if there exists 6j
such that
(26)
G9l(x-h)
lim ^ ,
* ? r - 7 T = + °°» f ° r a 1 1 h>0.
X _ H ^ G 9l (x) Log G 6l (x)
(If (26) istrue for 8 l9 then (26) is true for ail 8).
i As for theorem 3, put W* the intersection of 08* with the isobar containing X* . If
(W*)n is almost surely stable, then there exists a séquence (a,,),, such that for ail e > 0 :
P l L i m i n f ^ - e < W* <a n + e)} = 1.
Let i n be the isobar containing an. For ail e > 0, there exists (hjj)n and ( 1 ¾ two
séquences of isobars and T]>0 such that for ail 8 (19) holds. The inclusions (20) and
(21) show that under (H), (X*)n is almost surely stable if and only if (W*)n is almost
surely stable. Now, [Geffroy, 1958, 1959] gives (26) as a necessary and sufficient
condition for (W*)n to be almost surely stable, and this achieves the proof.
i
Let fe dénote the density of F Q .
COROLLARY
(27)
2.- If (H) isfulfilled, if there exists Qx such that
feiW
x^£o G ei (x) Log ILog G ei (x)l " + °°
then (X*)n is almost surely stable.
For the proof, see [Geffroy, 1958,1959].
STABILITY OF EXTREME VALUE
13
V - EXAMPLES
In this section, we suppose that k = 2 and we use the polar coordinates in E 2 .
1. EX AMPLE 1.- In this first example, Fe(x) = (1 - e" a ( e ) xm ) i {x > 0} where
m > 0 and a is a continuous, strictly positive function over [0,27c] such that
cc(0) = <x(27t). Then fe(x) = m a(8) x™-l e-<*(G)xm 1 {x < 0}.
Forafixed Qx and for every w > 0, the u(w)-level isobar g(8,w) isdefinedby
g(8,w) =
—
w.
a(8)J
Sothat (H) isfulfilled.
However (27) is satisfïed only for m > 1 since
fg(x)
G e (x) Log ILog Ge(x)l "
ma(8)xm-1
m
L o g ( tt (6).x ) *
Then (Xn)n is almost surely stable for m > 1.
2. EX AMPLE 2. Assume now that m = 1, so that Fô(x) = (1 - e-a(e)x) H {x > 0}
(exponential distribution). Then the conditions of Gnedenko's theorem are not satisfïed
and (Xn)n is not almost surely stable. As mentionned in the introduction, this suggests
a weaker notion than stability : the <p-stability. Before studying the q>-stability we examine
more deeply the condition (H).
VI - SOME REMARKS ABOUT (H).
First, it can be shown that (H) is not a necessary condition for the stability of
(X ) n ; for this, it suffices to consider the next example :
1. EXAMPLE 3.- Suppose that (X lt ...,X n ) is a sample form a common twodimensional distribution :
14
M.F. Delcroix et P. Jacob
Fô(x) = ( 1 - e - * a ( e ) ) ! { x > 0 } ,
(28)
where a is continuous over [0,2TU] , a(0) = OC(2TC) and a(8) > 0 for ail 8.
The isobars containing w > 0 are defined by :
q(Q)
g(8,w) = w«(ô) .
a) Clearly (H) is not fulfilled for this example. In fact, even (4) does not hold :
Choose <x(6) = n + 8 over [0,TC], then for 0 < 8 < TC,
1 ^ n
\ < —- < 1
(29)
7C+8
Suppose (4) true ; then for ail e > 0 there exists i\ > 0 and for ail w > 0 there exists
v(w) such that for ail 8 :
(30)
w*fc+e
< w^/Tt+e+T! < (vWf/n+Q <
n
y
/n+Q + e .
Particularly,
w < v(w) < w + e .
Therefore,
W^/TC+G
_ ( V (w))^ + Ô < w^+9 [(1 + £)*/*+e _ i]
and, for w large enough, for ail 8 > 0 ;
w%e.(v(w))%e
£ n
n,l.
< w*/*+e (^^L.
+ 0( ±))
W
w
7C+8
From (29) we deduce :
Lim
(
W^K+G -
(v(w))*/*+ô) = 0 ,
n-> +oo
and this contradicts (30).
b) However (Xn)n is almost surely stable.
STABILITY OF EXTREME VALUE
15
Let P(8) = l/<x(8) and suppose that P has a maximum P(0) = P < 1. The équation of
the isobars is :
(31)
g(e,w) =
WP
(e)/
P ;
since for ail 6 p(0) <. p < 1 ,
(32)
suplwP^-vP^l^w-vl
e
for w and v large enough, then
(33)
sup lg(8,w) - g(8,v)l = lg(0,w) - g(0,v)l
e
Now, put Wn the intersection of the isobar containing X n and the positive half-axis
8 = 0 ; (Wn)n is almost surely stable because (27) holds.
Hence, for ail e > 0 :
P {Liminf [ a n - e < W n <a lï + e]} = 1 ,
with ^ = 0^(^) = (Log n)P.
Let e > 0 and dénote by i" e (8)
(resp. ^ n (8), ijj(8)) the isobar containing ^- e
(resp. ^ , ^ + e). Since i" e and ijj are isobars,
1 = P {Liminf [a n -e<W* <an + e]}
= P {Lim inf f*(en)
<Rn<
i ^ ) }
Moreover from (33),
lim sup I i"n e (8) - i n (8)l = e
n->oo
G
and
lim sup l i n^ e ) - i n ( 8 ) l = e
n-*« e
16
M.F. Delcroix et P. Jacob
Then,
*
* Ps-
and (X ) n is almost surely stable.
2.- From this example we deduce another suffïcient condition for the stability of
(X ) n . Suppose there exists 8! such that for ail w > 0 and for ail T|>0
(K)
sup (g(8,w+7i) - g(8,w-Ti)) = gO^w+lD-gOl.w-Ti).
e
Let W be the intersection of the isobar containing Xfl and the positive half-axis OOj .
As for example 3 we can prove the next theorem.
*
*
THEOREM 5. / / (K) holds and if (W ) n is almost surely stable, then (X n ) n is
almost surely stable.
3.- Conditions (H) and (K) differ in kind : (H) is an uniformity condition whereas
(K) uses the existency of a direction 8^ which can be interpreted as the less favourable
direction.
Clearly (H) and (K) are not équivalent (see example 3). Of course if
g(8,w) = k(8)w + i ( 8 ) , then (H) involves (K). Remark also that under (K) theorem 3 is
still true.
VH - cp-STABILITY
l.-Let cp be a function over E + , positive, C1, increasing and bijective. In this
section we consider the variables Xj* = ( ¢ ^ ) , 0 ^ , . . , ^ = (¢(1^),0^.
Put for ail 8, f£ the conditional density of cp(R) given 0 = 8 and F^ = 1 - G^ the
conditional distribution function of (p(R) given 0 = 8.
For ail t >0,
(34)
F£(t) = Fe((p-l(t))
STABILITY OF EXTREME VALUE
fj(t) = (cp-i(t))'. f9(cp-1(0).
(35)
Put
17
X** = ((cp(Rn))*,©ï;)
the maximum value of
(X\
X^) ; then,
Xj* = (cp(R*),0*).
Clearly, if (gn)n is a séquence of u-level isobars for the distribution of (R,0), then
(<p(gn))n is a séquence of u-level isobars for the distribution of (<p(R),0) and
conversely.
This remark entails a définition which generalizes the définition of relative stability
given by Geffroy in [Geffroy, 1958,1959] and Gnedenko in [Gnedenko, 1943].
DÉFINITION
3.- The séquence (X ) n is called (p-stable in probability (almost
surely) if there exists a séquence of isobars (gn)n (for the distribution of (R,0)) such
that:
(36)
cp(R*)-<p(gn(0*)) -* 0.
a.s.
REMARJC
6.- When <p(x) = Log (x) = Max(0,Logx), <p -stabilityreducesto relative
stability [Geffroy, 1958,1959], [Gnedenko, 1943], [Green,1976] ; (36) can be written :
(37)
-&-
-^ 1
2.- AU the results of sections ni and IV can be used for (X^XΣf...,X^) :
a) It is easily seen that the équation of the isobar containing v > 0 on the
half-axis 0Ql is :
(38)
gne,v) = (p(g(6,<p-1(v)))
b) Condition (H) becomes (H)<P ; for example, if <p = Log, (H)<P reduces
to:
18
M.F. Delcroix et P. Jacob
9g(9,w)
(H)
a, <
1
^ — < p,
g(6,w)
Hl
c) Theorems 3 and 4 can be used with G^, f£ ,... instead of G^ , f0.
*
<p*
d)Of course, if (X ) n is stable, stability properties subsit for (X ) n if
(p is a concave function, [Gather and Rauhut, 1990].
3 . - EXAMPLES
a) Example 4 : Cauchy's distribution.
Suppose E n is a sample from a multidimensional distribution :
fQ0 (x) = - ,Mel 2
l{x>0} ;
7C x2 + X (6)
X is a continuous and positive function.
Then
Fe(x) = ?-Arctg ( - 7 - ) M x > 0 ) .
K
X(0)
*
Since Cauchy's distribution does not hâve moment of any order, (X n ) n is not stable
[Geffroy, 1958, 19591.
However, for ail 9 and for x large enough,
Fe(x) « 1 - - - 3x - je
Then the conditions of Gnedenko's theorem are satisfïed by G? if and only if :
(39)
lim
^ (x"h)
X-> +oo
(P _1 (X)
= 0> f o r a l l
h > 0
STABILITY OF EXTREME VALUE
19
It suffices to choose cp(x) = VLogx or cp(x) = Log Log x which is more concave than
VLogx. Moreover for this two functions, (H*P) holds for v large enough. Therefore
(X ) n is <p-stable in probability. Note that
9 a ( x ) = (Log x ) a ,
(0 < a < 1) is also a suitable function, but (X ) n is not relatively stable.
b) Example 5 : exponential distribution.
Suppose En is a sample from a multidimensional distribution :
Fe(x) = ( l - e-a(e)x) 1 {x>0} ;
a is a continuous and positive function.
Similarly, for an exponential distribution, the condition for q> is :
(40)
lim (p-1(x-h)-<p"1(x) = -«», for ail h > 0 .
x-> +«°
>\
The functions cp(x) = Vx or <p(x) = Log (x) are suitable and (H)<P is also satisfïed.
c) Example 6.
More generally, consider Example 1 with 0 < m < 1. Choose (p(x) =
xl/2ms
^en F^(x) = (1 -e"0*9)*2) i {x>0} and (X9*)
is almost surely stable, as
it has been shown in Example 1.
d) Example 7.
Suppose En is a sample from a multidimensional distribution.
X and a are continuous and positive functions. (For a (6) = 1, we obtain
asymptotically a Cauchy's distribution). Choose <p(x) = VLogx ; then ,
F%) = (1-^(9) e- a ( e ) x2 ) i {x>0}
9
and Example 1 shows that (X ) is almost surely stable.
20
M.F. Delcroix et P. Jacob
V i n - CONCLUDING REMARKS
In view of the previous results, we are led to consider the géométrie aspect of the
stability : in a further paper we plan to describe how the sample set of points lies in a
shape limited by the isobars yn . Geffroy [1958,19591, Fisher [1966] were thefirstto
deal with this subject in the case of a convex shape (see also [Davis, Mulrow, Resnick,
19871), as for the typical example of a gaussian sample. Using the isobars it is possible
to study the case of a non necessarily convex shape.
Moreover it would be interesting to also examine the statistical aspect of the
problem : for example to give a functional estimate of yn for an unknown sample.
STABILITY OF EXTREME VALUE
21
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